A conjugate bivariate model suitable for detecting heterogeneity and getting evaluation marks via multiple-choice tests

Emilio Gómez-Déniz; Nancy Dávila-Cárdenes; José María Pérez-Sánchez; Emilio Gómez-Déniz; Nancy Dávila-Cárdenes; José María Pérez-Sánchez

doi:10.3934/math.2026819

AIMS Mathematics

2026, Volume 11, Issue 7: 20170-20194. doi: 10.3934/math.2026819

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Research article

A conjugate bivariate model suitable for detecting heterogeneity and getting evaluation marks via multiple-choice tests

1.
Department of Quantitative Methods in Economics and TiDES Institute, University of Las Palmas de Gran Canaria, Spain
2.
Department of Applied Economic Analysis and TiDES Institute, University of Las Palmas de Gran Canaria, Spain

Received: 18 May 2026 Revised: 22 June 2026 Accepted: 29 June 2026 Published: 09 July 2026
MSC : 97Mxx, 97Rxx

It is well known that academic performance varies in heterogeneous and homogeneous groups. In the latter, all students possess approximately the same level of instruction and skills. The results are usually measured by the marks achieved on exams. In this work, the marks calculated through this methodology could be considered a Bayes rule within the framework of statistical decision theory. With this methodology, we estimated the marks obtained on a multiple-choice test, considering individual student performance with the accumulated experience of the group. As a result, we addressed two major objectives: First, a mechanism was developed to detect the heterogeneity or homogeneity of the group based on the accumulated experience on past exams; and second, an evaluation system was proposed that rewarded or penalized students depending on their academic performance. The proposed framework may have implications for supporting students at risk, although such effects were not directly evaluated in this study.
- Bayesian,
- classroom,
- conjugate distribution,
- heterogeneity,
- loss function,
- marks,
- multiple-choice test
Citation: Emilio Gómez-Déniz, Nancy Dávila-Cárdenes, José María Pérez-Sánchez. A conjugate bivariate model suitable for detecting heterogeneity and getting evaluation marks via multiple-choice tests[J]. AIMS Mathematics, 2026, 11(7): 20170-20194. doi: 10.3934/math.2026819

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Abstract

It is well known that academic performance varies in heterogeneous and homogeneous groups. In the latter, all students possess approximately the same level of instruction and skills. The results are usually measured by the marks achieved on exams. In this work, the marks calculated through this methodology could be considered a Bayes rule within the framework of statistical decision theory. With this methodology, we estimated the marks obtained on a multiple-choice test, considering individual student performance with the accumulated experience of the group. As a result, we addressed two major objectives: First, a mechanism was developed to detect the heterogeneity or homogeneity of the group based on the accumulated experience on past exams; and second, an evaluation system was proposed that rewarded or penalized students depending on their academic performance. The proposed framework may have implications for supporting students at risk, although such effects were not directly evaluated in this study.

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